50. This week, Boya has 20 rocks collected, and her goal is to have a total of 160 rocks collected 8 weeks from now. After adding to her rock collection next week, each week she will add 1 more rock than what she added the previous week. For the next 8 weeks, Boya will not remove any rocks from what she has already collected. What is the minimum number of rocks Boya must add to her collection next week so that she reaches her goal?

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Let $$ x $$ be the number of rocks Boya adds to her collection next week. According to the problem, she will add rocks in the following manner:

- In week 1, she adds $$ x $$ rocks.
- In week 2, she adds $$ x + 1 $$ rocks.
- In week 3, she adds $$ x + 2 $$ rocks.
- ...
- In week 8, she adds $$ x + 7 $$ rocks.

The total number of rocks added over the 8 weeks can be calculated as follows:

$$
\text{Total rocks added} = x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6) + (x + 7)
$$

This simplifies to:

$$
\text{Total rocks added} = 8x + (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7)
$$

The sum of the integers from 0 to 7 is:

$$
0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = \frac{7 \cdot (7 + 1)}{2} = \frac{7 \cdot 8}{2} = 28
$$

Thus, the total number of rocks added over the 8 weeks is:

$$
\text{Total rocks added} = 8x + 28
$$

Boya currently has 20 rocks and wants to reach a total of 160 rocks. Therefore, the equation for the total number of rocks after 8 weeks is:

$$
20 + (8x + 28) = 160
$$

Simplifying this equation gives:

$$
20 + 8x + 28 = 160
$$
$$
8x + 48 = 160
$$
$$
8x = 160 - 48
$$
$$
8x = 112
$$
$$
x = \frac{112}{8} = 14
$$

Thus, the minimum number of rocks Boya must add to her collection next week is:

$$
\boxed{14}
$$
Boya needs to add 14 rocks next week to reach her goal of 160 rocks in 8 weeks.

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